where \(\text{prox}_{h^{*}_{f}}(\mathbf {u})=\arg \min \limits _{\mathbf {y}}||\mathbf {y}-\mathbf {u}||_F^2+h^{*}_{f}(\mathbf {u})\) . Similar to “predictor-corrector" algorithm [1]}, we adopt iterative algorithm to find saddle point for min-max optimization problem. Specifically, starting from \((\mathbf {F}^{k}, \mathbf {u}^{k})\) , we adopt a gradient descent step on primal variable \(\mathbf {F}\)  to arrive \((\bar{\mathbf {F}}^{k+1}, \mathbf {u}^{k})\)  and then followed by a proximal ascent step in the dual variable \(\mathbf {u}\) . Finally, a gradient descent step on primal variable in point \((\bar{\mathbf {F}}^{k+1}, \mathbf {u}^{k})\)  to arrive at \((\mathbf {F}^{k+1}, \mathbf {u}^{k})\) . In short, the iteration can be summarized as
\(\left\lbrace \begin{array}{l}\bar{\mathbf {F}}^{k+1}=\mathbf {F}^{k}-\gamma \nabla h_s(\mathbf {F}^{k})-\gamma \frac{\partial \langle \mathbf {p}, \mathbf {u}^{k}\rangle }{\partial \mathbf {F}}\Big |_{\mathbf {F}^{k}}, \\\mathbf {u}^{k+1} = \text{prox}_{\beta h^{*}_{f}}\big (\mathbf {u}^{k}+\beta \mathbf {\Delta _{s}} SF(\bar{\mathbf {F}}^{k+1})\big ), \\\bar{\mathbf {F}}^{k+1}=\mathbf {F}^{k}-\gamma \nabla h_s(\mathbf {F}^{k})-\gamma \frac{\partial \langle \mathbf {p}, \mathbf {u}^{k+1}\rangle }{\partial \mathbf {F}}\Big |_{\mathbf {F}^{k}}. \\\end{array}\right.\) 
